The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.
The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.
We investigate the low-temperature phase of the two-dimensional dipolar XY model. Within the framework of the nonlinear $\ensuremath{\sigma}$ model we show that the dipolar XY model is renormalizable, despite the anisotropic character of the dipolar interaction. We predict a low-temperature phase with spontaneous magnetization governed by a low-temperature fixed point with infinite dipolar coupling. The critical fixed point shows nonlinear behavior, leading to unusual critical phenomena; e.g., the correlation length and spontaneous magnetization show exponential behavior instead of power laws. The nature and flow diagram of the ferromagnetic transition are strikingly similar to the Kosterlitz-Thouless transition. But while in the Kosterlitz-Thouless transition the exponential behavior is a consequence of the topological excitations, the predicted phenomena in the dipolar XY model are solely due to spin-wave excitations. In addition to the dXY model we introduce a whole class of long-range XY models showing such nonstandard behavior. Parametrizing the divergences in terms of the correlation length we calculate the critical exponents of the transition. These exponents are correct in any loop order. The influence of topological excitations on the ferromagnetic transition is discussed.
Abstract The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of LR interacting systems. The competition between LR coupling and the Berezinskii–Kosterlitz–Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction ∼1/r 2+σ . Utilizing an advanced update strategy, we conduct LR Monte Carlo simulations of the model up to a linear size of L = 8192. Our results demonstrate continuous phase transitions into a ferromagnetic phase for σ < 2, which exhibit the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at σ = 2. The observed scaling behaviors in the low-temperature phase for σ ≤ 2 agree with our theoretical analysis. Our findings request further theoretical understanding and can be of practical application in cutting-edge experiments like Rydberg atom arrays.
We discuss the critical behaviour of the 1-dimensional classical XY model with power law decay long range interactions by using Monte Carlo simulation. It has been predicted by renormalization group calculation that the critical behaviour changes with the power of interactions. From the simulation result of temperature dependence of the specific heat, we confirm that the system shows the critical behaviour of the mean field type and the non-trivial long range order. We also estimate the critical exponent η for the system with the inverse square long range interactions, and by using the phenomenological renormalization group plot, we confirm the 'Berezinskii-Kosterlitz-Thouless (BKT)-like' transition which has been predicted by the spin wave calculations.
In this work we have used extensive Monte Carlo simulations and finite size scaling theory to study the phase transition in the dipolar planar rotator model (dPRM), also known as dipolar XY model. The true long-range character of the dipolar interactions was taken into account by using the Ewald summation technique. Our results for the critical exponents do not fit those from known universality classes. We observed that the specific heat is apparently non-divergent and the critical exponents are ν = 1.277(2), β = 0.2065(4) and γ = 2.218(5). The critical temperature was found to be T(c) = 1.201(1). Our results are clearly distinct from those of a recent renormalization group study from Maier and Schwabl (2004 Phys. Rev. B 70 134430) and agrees with the results from a previous study of the anisotropic Heisenberg model with dipolar interactions in a bilayer system using a cut-off in the dipolar interactions (Mól and Costa 2009 Phys. Rev. B 79 054404).
The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions $\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}$ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $\ensuremath{\sigma}\ensuremath{\le}2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for $\ensuremath{\sigma}<2$, with decaying exponent $2\ensuremath{-}\ensuremath{\sigma}$, and in the form of the inverse logarithm of distance for $\ensuremath{\sigma}=2$. Moreover, the critical points and exponents are also determined for various $\ensuremath{\sigma}$. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at $\ensuremath{\sigma}=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.
Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XY Model with 1/r4 Interaction
We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as 1/r2+σ. At the marginal case σ=2, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, C(r)∼(lnr)−ηℓ with ηℓ∝T, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the 1/r4 interaction logarithmically modifies the spin-wave stiffness kernel as γ(k)≃κk2ln(1/k), which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent ν∝1/2πκ. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for σ≤2. These results establish σ=2 as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.